21Hint : trigonometry
A nice problem ;) (who am i to rate ? :P)
If u choose 5 distinct real numbers there must exist some a and b for which the following inequality is valid
|ab+1| > |a-b|
Prove it!!
source : ISI workbook
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4 Answers
21hmm..Yes it can be said , but when i did it was plain common sense ,i m talking abt the PHP part ;
21solution ;)
write the given inequality as 1> |a-b|/|ab+1|
or 1> |a-bab+1|
tan is an increasing function in the interval (-∩/2 , ∩/2) and so is tan-1.
so the given inequality becomes tan-11 > tan-1|a-bab+1|
which gives a hint to use Co ordinate geometry,we recall that the slope of a line can be any real number.So we represent the selected real number(say X) by a line passing through the origin with a slope X.obviously we will get distinct lines, because in the interval (0 ,∩) tan is a bijection
therefore, tan-1|a-bab+1| represents the acute angle between such lines.
If there are 5 such lines , its obvious * that the acute angle between at least two of them is less than 450
* http://en.wikipedia.org/wiki/Pigeonhole_principle